On the 3-dimensional fractional-order Toda lattice with two controls

The paper studies a 3D fractional-order Toda lattice (with and without controls), applies Matignon’s criterion to the Jacobian at equilibria, and derives sign conditions for (a)symptotic stability. At the two families of equilibria considered, the Jacobians are diagonal, so the eigenvalues (and thus the stability verdicts) follow immediately from their signs; this is formalized in Proposition 3.1 for e0 and Proposition 3.2 for ek,m23, and the candidate solution reproduces exactly this computation and these conditions . However, both the paper and the model invoke a blanket rule that any zero eigenvalue implies instability for all q∈(0,1) (Corollary 2.2(i)), which is not a consequence of Matignon’s theorem and is in fact false for the linear test equation D^q x=0 (Lyapunov stable but not asymptotically stable). The paper’s Proposition 2.4 concludes that the uncontrolled equilibria are “unstable for all q” solely from the presence of zero eigenvalues, which overreaches what is proved; the justified conclusion is merely “not asymptotically stable” unless additional nonlinear analysis is supplied . Thus, while the eigenvalue calculations and controlled-case criteria are correct and aligned in both sources, both are incomplete in their handling of zero eigenvalues in the uncontrolled case.